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Fourier analysis for discontinuous Galerkin and related methods

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chinese Science Bulletin 2009年第11期54卷 1809-1816页

作者： ZHANG MengPing shu chi-wang Department of Mathematics University of Science and Technology of China Hefei 230026 China Division of Applied Mathematics Brown University Providence RI 02912 USA

In this paper we review a series of recent work on using a Fourier analysis technique to study the sta-bility and error estimates for the discontinuous Galerkin method and other related schemes. The ad-vantage of this approach is that it can reveal instability of certain "bad" ' schemes; it can verify stability for certain good schemes which are not easily amendable to standard finite element stability analysis techniques; it can provide quantitative error comparisons among different schemes; and it can be used to study superconvergence and time evolution of errors for the discontinuous Galerkin method. We will briefly describe this Fourier analysis technique, summarize its usage in stability and error estimates for various schemes, and indicate the advantages and disadvantages of this technique in comparison with other finite element techniques.

关键词：间断Galerkin方法傅里叶分析相关方有限元技术稳定性分析傅立叶分析误差估计时间演化

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

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Science china Mathematics 2013年第12期56卷 2645-2676页

作者： LU JianFang shu chi-wang ZHANG MengPing School of Mathematical Sciences University of Science and Technology of China Division of Applied Mathematics Brown UniversityProvidence RI 02912 USA

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in *** RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical *** using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth *** theoretical results are proved for piecewise polynomials with any degree k ***,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising *** results are provided to demonstrate these RKDG methods.

关键词： discontinuous Galerkin method explicit Runge-Kutta method stability error estimates corre-lated random walk positivity-preserving

A Macroscopic Approach to the Lane Formation Phenomenon in Pedestrian Counterflow

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chinese Physics Letters 2011年第10期28卷 277-280页

作者： XIONG Tao ZHANG Peng WONG SC shu chi-wang ZHANG Meng-Ping Department of Mathematics University of Science and Technology of ChinaHefei 230026 Shanghai Institute of Applied Mathematics and Mechanics Shanghai UniversityShanghai 200072 Department of Civil Engineering The University of Hong KongHong Kong Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We simulate pedestrian counterflow by adopting an optimal path-choice strategy and a recently observed speed-density *** the whole system is symmetric,the simulation demonstrates the segregation and formation of many walking lanes for two groups of *** symmetry breaking is most likely triggered by a small numerical viscosity or"noise",and the segregation is associated with the minimization of travel *** underlying physics can be compared with the"optimal self-organization"mechanism in Helbing's social force model,by which driven entities in an open system tend to minimize their interaction to enable them to reach some ordering state.

关键词： flow system optimal

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爆炸力学高精度数值模拟研究进展

科学通报 2015年第10期60卷 882-898页

作者：王成 shu chi-wang 北京理工大学爆炸科学与技术国家重点实验室 Division of Applied Mathematics Brown UniversityProvidence RI 02912USA

爆炸问题的高精度数值模拟在武器弹药设计及毁伤评估、工业重大爆炸灾害等国防和民用问题中具有重要的理论价值和应用前景.本文对近十几年来爆炸力学高精度计算的研究作了回顾与评述,结合作者近年来的研究工作,着重总结了高精度数值模... 详细信息

爆炸问题的高精度数值模拟在武器弹药设计及毁伤评估、工业重大爆炸灾害等国防和民用问题中具有重要的理论价值和应用前景.本文对近十几年来爆炸力学高精度计算的研究作了回顾与评述,结合作者近年来的研究工作,着重总结了高精度数值模拟保正性、高精度边界条件、自适应网格技术和多物质界面处理等方面的研究进展.推广并发展了WENO格式和RKDG格式,构造了守恒型高精度保正计算格式,在复杂爆轰波问题的数值模拟中很好地解决了出现负密度和负压强的问题,清晰地捕捉了爆轰波阵面结构和流动特征;提出了基于笛卡尔网格的高精度Inverse Lax-Wendroff复杂边界处理方法,该方法通过Inverse Lax-Wendroff获得一阶的法向偏导数值,所有其他的高阶法向偏导数值由五阶WENO类型的外插方法获得,实现了含复杂边界爆轰问题的高精度数值模拟,并具有较好的稳定性和鲁棒性;将后验误差估计方法应用于求解Euler方程中,采用网格的后验误差作为细分的判据,网格细分时在空间上利用高阶WENO插值,时间上采用Hermite插值.对于不需要细分的网格,本文采用点对点的直接赋值方法进行合并;发展了多介质界面处理的Local Level Set方法,对计算域内速度场进行修正,有效避免了速度场间断在求解界面位置时引起的误差;将GFM(ghost fluid method)和RGFM(real ghost fluid method)界面处理方法相结合,避免了GFM方法处理物质界面强间断时引入的虚假物理解,克服了RGFM方法处理弱间断时,重复求解Riemann问题引起的计算资源浪费.最后,对爆炸力学高精度计算的发展前沿进行了展望.

关键词：爆炸力学高精度数值模拟多物质界面并行计算

Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

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Communications on Applied Mathematics and Computation 2024年第2期6卷 907-962页

作者： Dinshaw S.Balsara Deepak Bhoriya chi-wang shu Harish Kumar Physics Department University of Notre DameNotre DameUSA ACMS Department University of Notre DameNotre DameUSA Division of Applied Mathematics Brown UniversityProvidenceUSA Department of Mathematics Indian Institute of TechnologyDelhiIndia

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation *** multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable *** makes them quite attractive for several science and engineering ***,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative *** this paper,we perform such an extension which improves the domain of the applicability of such *** extension is carried out by writing the scheme in fluctuation *** use the HLLI Riemann solver of Dumbser and Balsara(***.304:275-319,2016)as a building block for carrying out this *** of the use of an HLL building block,the resulting scheme has a proper supersonic *** use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the *** new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present *** conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied *** examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional *** Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the *** some

关键词： Hyperbolic PDEs Numerical schemes Non-conservative products Stiff source terms Finite difference WENO

Error analysis of the semi-discrete local discontinuous Galerkin method for semiconductor device simulation models

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Science china Mathematics 2010年第12期53卷 3255-3278页

作者： LIU YunXian 1, & shu chi-wang 2 1 School of Mathematics, Shandong University, Jinan 250100, china 2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA School of Mathematics Shandong University Jinan China Division of Applied Mathematics Brown University Providence USA

In this paper we continue our effort in Liu-shu (2004) and Liu-shu (2007) for developing local discontinuous Galerkin (LDG) finite element methods to discretize moment models in semiconductor device simulations. We consider drift-diffusion (DD) and high-field (HF) models of one-dimensional devices, which involve not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. Error estimates are obtained for both models with smooth solutions. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. A simulation is also performed to validate the analysis.

关键词： local discontinuous Galerkin method error estimate semiconductor

计算流体力学中的高精度数值方法回顾(英文)

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计算物理 2009年第5期26卷 633-655页

作者：成娟舒其望北京应用物理与计算数学研究所北京100088 布朗大学美国罗德岛州普罗维登斯02912

在过去的二、三十年中,计算流体力学(CFD)领域的高精度数值方法的设计和应用研究非常活跃.高精度数值方法主要针对具有复杂解结构流场的模拟而设计.回顾CFD中主要用于可压缩流模拟的几类高精度格式的发展与应用.可压缩流的一个重要特征... 详细信息

在过去的二、三十年中,计算流体力学(CFD)领域的高精度数值方法的设计和应用研究非常活跃.高精度数值方法主要针对具有复杂解结构流场的模拟而设计.回顾CFD中主要用于可压缩流模拟的几类高精度格式的发展与应用.可压缩流的一个重要特征是流场中存在激波、界面以及其它间断,同时还常常在解的光滑区域包含复杂结构.这对设计既不振荡又保持高阶精度的格式带来特别的挑战.重点讨论本质无振荡(ENO)、加权本质无振荡(WENO)有限差分与有限体积格式、间断Galerkin有限元(DG)方法,描述它们各自的特点、长处与不足,简要回顾这些方法的发展和应用,重点介绍它们近五年来的最新进展.

关键词：本质无振荡(ENO) 加权本质无振荡(WENO) 间断Galerkin(DG) 高精度有限差分有限体积有限元计算流体力学可压缩流

Energy-Function Based Pilot Protection Scheme for Hybrid UHVDC Transmission Applying Improved Hausdorff Distance

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CSEE Journal of Power and Energy Systems 2024年第3期10卷 891-902页

作者： Zijiang wang Youping Fan Ben Shang Yinbiao shu School of Electrical Engineering and Automation Wuhan UniversityWuhan 430072China Institute of Next Generation Power Systems&International Standards Wuhan UniversityWuhan 430072China CSEE IEEE

A hybrid UHVDC transmission system applying LCC as the rectifier and MMC as the inverter combines the advantages of both converter types,which makes this protection scheme more complicated.A new pilot protection scheme for a three-terminal hybrid DC transmission system applying energy functions is *** energy function for LCC is applied to MMC to derive the energy level of the hybrid ***,an improved Hausdorff distance(IHD)algorithm is proposed to detect the difference in energy levels between the normal and fault *** abrupt change in energy level is characterized by IHD change *** points at which the IHD change rate exceeds the threshold at converter stations are applied to determine the fault line and to estimate the fault *** proposed protection scheme is then verified by a simulation model of the Wudongde±800 kV three-terminal hybrid UHVDC transmission *** appropriate sampling frequency is selected for a real-time calculation,and the threshold is selected considering the effect of *** show the proposed scheme can identify and trip fault lines quickly and effectively,even for a 600Ωgrounding *** waveshape similarity algorithms are compared and *** with existing protection schemes,the proposed scheme transmits less data to improve communication speed and reliability.

关键词： Energy function Hausdorff distance pilot protection UHVDC transmission

Verification and Validation of High-Resolution Inviscid and Viscous Conical Nozzle Flows

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Communications on Applied Mathematics and Computation 2024年第1期6卷 533-549页

作者： Luciano K.Araki Rafael B.de R.Borges Nicholas Dicati P.da Silva chi-wang shu Department of Mechanical Engineering Federal University of ParanáCuritiba81531-980ParanáBrazil Mathematics and Statistics Institute Rio de Janeiro State UniversityRio de Janeiro20550-900Rio de JaneiroBrazil Department of Mechanical Engineering State University of MaringáMaringá87020-900ParanáBrazil Division of Applied Mathematics Brown UniversityProvidenceRI02912USA

Capturing elaborated flow structures and phenomena is required for well-solved numerical *** finite difference methods allow simple discretization of mesh and model ***,they need simpler meshes,e.g.,*** inverse Lax-Wendroff(ILW)procedure can handle complex geometries for rectangular ***-resolution and high-order methods can capture elaborated flow structures and *** also have strong mathematical and physical backgrounds,such as positivity-preserving,jump conditions,and wave propagation *** perceive an effort toward direct numerical simulation,for instance,regarding weighted essentially non-oscillatory(WENO)***,we propose to solve a challenging engineering application without turbulence *** aim to verify and validate recent high-resolution and high-order *** check the solver accuracy,we solved vortex and Couette ***,we solved inviscid and viscous nozzle flows for a conical *** employed the finite difference method,positivity-preserving Lax-Friedrichs splitting,high-resolution viscous terms discretization,fifth-order multi-resolution WENO,ILW,and third-order strong stability preserving *** showed the solver is high-order and captured elaborated flow structures and *** can see oblique shocks in both nozzle *** the viscous flow,we also captured a free-shock separation,recirculation,entrainment region,Mach disk,and the diamond-shaped pattern of nozzle flows.

关键词： High-resolution Compressible Navier-Stokes Free-shock separation Nozzle flow